Modelling decaying seasonal variation with SARIMA I have a time series on quarterly data with seasonality which I am trying to fit a SARIMA model to. The seasonal variation seems to be decreasing with time. I am wondering what the best way to model this with a SARIMA model is. Here is the series decomposed: 
My confusion stems from when it is appropriate to take the first seasonal difference. In the literature, taking the first seasonal difference seems to be the correct way to handle seasonality. But if I do that, I model that the seasonal difference is constant over time, which it doesn't seem to be. If I then try to model without taking the first seasonal difference, all reasonable models violate the assumption of the AR constants being between -1 and 1. Here are the four quarters plotted separately:

 A: 
My confusion stems from when it is appropriate to take the first seasonal difference. In the literature, taking the first seasonal difference seems to be the correct way to handle seasonality.

Seasonal differencing should be applied when the series is seasonally integrated. That happens when the time series is generated from alternating random walks (one per period). As a consequence, any two consecutive observations of the time series diverge (!) under seasonal integration, which is often hard to justify. Thus I would think twice before applying seasonal differencing. 
Regarding diminishing seasonality, I think seasonal AR or MA terms in the SARIMA model might work fine. But I am not quite sure. You could try auto.arima in R to see what SARIMA model is suggested and how the model residuals look; perhaps they will be adequate.
A: If you have strong feeling (and some theoretical foundation) that your time series have and will have decreasing seasonality, I would suggest trying two options:
1) Transform the time series (for instance by taking logarithm log(x))
2) Opt for second degree of seasonal differencing (D=2)
